Chapter 10: Fourier Transform Properties

The Discrete Time Fourier Transform

The Discrete Time Fourier Transform (DTFT) is the member of the Fourier
transform family that operates on aperiodic, discrete signals. The best way to
understand the DTFT is how it relates to the DFT. To start, imagine that you
acquire an N sample signal, and want to find its frequency spectrum. By using
the DFT, the signal can be decomposed into sine and cosine waves, with
frequencies equally spaced between zero and one-half of the sampling rate.
As discussed in the last chapter, padding the time domain signal with zeros
makes the period of the time domain longer, as well as making the spacing
between samples in the frequency domain narrower. As N approaches infinity,
the time domain becomes aperiodic, and the frequency domain becomes a
continuous signal. This is the DTFT, the Fourier transform that relates an
aperiodic, discrete signal, with a periodic, continuous frequency spectrum.

The mathematics of the DTFT can be understood by starting with the synthesis
and analysis equations for the DFT (Eqs. 8-2, 8-3 and 8-4), and taking N to
infinity:

There are many subtle details in these relations. First, the time domain signal, x[n], is still discrete, and therefore is represented by brackets. In comparison, the frequency domain signals, ReX(ω) & ImX(ω), are continuous, and are thus
written with parentheses. Since the frequency domain is continuous, the
synthesis equation must be written as an integral, rather than a summation.

As discussed in Chapter 8, frequency is represented in the DFT's frequency
domain by one of three variables: k, an index that runs from 0 to N/2; f, the
fraction of the sampling rate, running from 0 to 0.5; or ω, the fraction of the
sampling rate expressed as a natural frequency, running from 0 to π. The
spectrum of the DTFT is continuous, so either f or ω can be used. The
common choice is ω, because it makes the equations shorter by eliminating the
always present factor of 2π. Remember, when ω is used, the frequency
spectrum extends from 0 to π, which corresponds to DC to one-half of the
sampling rate. To make things even more complicated, many authors use Ω (an
upper case omega) to represent this frequency in the DTFT, rather than ω (a
lower case omega).

When calculating the inverse DFT, samples 0 and N/2 must be divided by two
(Eq. 8-3) before the synthesis can be carried out (Eq. 8-2). This is not necessary
with the DTFT. As you recall, this action in the DFT is related to the frequency
spectrum being defined as a spectral density, i.e., amplitude per unit of
bandwidth. When the spectrum becomes continuous, the special treatment of
the end points disappear. However, there is still a normalization factor that
must be included, the 2/N in the DFT (Eq. 8-3) becomes 1/π in the DTFT (Eq.
10-2). Some authors place these terms in front of the synthesis equation, while
others place them in front of the analysis equation. Suppose you start with
some time domain signal. After taking the Fourier transform, and then the
Inverse Fourier transform, you want to end up with what you started. That is,
the 1/π term (or the 2/N term) must be encountered somewhere along the way,
either in the forward or in the inverse transform. Some authors even split the
term between the two transforms by placing 1/√π in front of both.

Since the DTFT involves infinite summations and integrals, it cannot be
calculated with a digital computer. Its main use is in theoretical problems as an
alternative to the DFT. For instance, suppose you want to find the frequency
response of a system from its impulse response. If the impulse response is
known as an array of numbers, such as might be obtained from an experimental
measurement or computer simulation, a DFT program is run on a computer.
This provides the frequency spectrum as another array of numbers, equally
spaced between 0 and 0.5 of the sampling rate.

In other cases, the impulse response might be know as an equation, such as a
sinc function or an exponentially decaying sinusoid. The DTFT is used here to
mathematically calculate the frequency domain as another equation, specifying
the entire continuous curve between 0 and 0.5. While the DFT could also be
used for this calculation, it would only provide an equation for samples of the
frequency response, not the entire curve.